On Microscopic Origin of Integrability in Seiberg-Witten Theory

We discuss microscopic origin of integrability in Seiberg-Witten theory, following mostly the results of hep-th/0612019, as well as present their certain extension and consider several explicit examples. In particular, we discuss in more detail the theory with the only switched on higher perturbation in the ultraviolet, where extra explicit formulas are obtained using bosonization and elliptic uniformization of the spectral curve.Comment: 24 pages, 1 figure, LaTeX, based on the talks at 'Geometry and Integrability in Mathematical Physics', Moscow, May 2006; 'Quarks-2006', Repino, May 2006; Twente conference on Lie groups, December 2006 and 'Classical and Quantum Integrable Models', Dubna, January 200

Glueball operators and the microscopic approach to N=1 gauge theories

We explain how to generalize Nekrasov's microscopic approach to N=2 gauge theories to the N=1 case, focusing on the typical example of the U(N) theory with one adjoint chiral multiplet X and an arbitrary polynomial tree-level superpotential Tr W(X). We provide a detailed analysis of the generalized glueball operators and a non-perturbative discussion of the Dijkgraaf-Vafa matrix model and of the generalized Konishi anomaly equations. We compute in particular the non-trivial quantum corrections to the Virasoro operators and algebra that generate these equations. We have performed explicit calculations up to two instantons, that involve the next-to-leading order corrections in Nekrasov's Omega-background.Comment: 38 pages, 1 figure and 1 appendix included; v2: typos and the list of references corrected, version to appear in JHE

Matone's Relation in the Presence of Gravitational Couplings

The prepotential in N=2 SUSY Yang-Mills theories enjoys remarkable properties. One of the most interesting is its relation to the coordinate on the quantum moduli space $u=$ that results into recursion equations for the coefficients of the prepotential due to instantons. In this work we show, with an explicit multi-instanton computation, that this relation holds true at arbitrary winding numbers. Even more interestingly we show that its validity extends to the case in which gravitational corrections are taken into account if the correlators are suitably modified. These results apply also to the cases in which matter in the fundamental and in the adjoint is included. We also check that the expressions we find satisfy the chiral ring relations for the gauge case and compute the first gravitational correction.Comment: 21 page

Aspects for efficient wide spectral band THz generation via CO2 laser down conversion

Detailed model study of THz generation by CO2 laser down-conversion in pure and solid solution crystals GaSe1-xSx is carried out for the first time. Both forward and backward collinear interactions of common (eo-e, oe-e, oe-o, oo-e, ee-o) and original (ee-e, oo-o) types are considered. Possibility of realization, phase matching angles and figure of merits are estimated for line mixing within 9 μm and 10 μm emission bands, as well between them. Dispersion properties of o- and e-wave refractive indices and absorption coefficients for GaSe, GaS and GaSe1-xSx crystals were preliminary measured by THz-TDS, approximated in the equation form and then used in the study. Estimated results are presented in the form of 3-D figures that are suitable for rapid analyses of DFG parameters. The most efficient type of interaction is eo-o type. Optimally doped (x = 0.09-0.13) GaSe1-xSx crystals are from 4 to 5 times more efficient at limit pump intensity than not doped GaSe crystals

On the Chiral Ring of N=1 Supersymmetric Gauge Theories

We consider the chiral ring of the pure N=1 supersymmetric gauge theory with SU(N) gauge group and show that the classical relation S^{N^2}=0 is modified to the exact quantum relation (S^N-\Lambda^{3N})^N=0.Comment: 5 pages. Comments and references adde

N=1 Superpotentials from Multi-Instanton Calculus

In this paper we compute gaugino and scalar condensates in N=1 supersymmetric gauge theories with and without massive adjoint matter, using localization formulae over the multi--instanton moduli space. Furthermore we compute the chiral ring relations among the correlators of the $N=1^*$ theory and check this result against the multi-instanton computation finding agreement.Comment: 31 pages, uses youngtab.sty, some explanations added, version to appear in JHE

Twisted supersymmetric 5D Yang-Mills theory and contact geometry

We extend the localization calculation of the 3D Chern-Simons partition function over Seifert manifolds to an analogous calculation in five dimensions. We construct a twisted version of N=1 supersymmetric Yang-Mills theory defined on a circle bundle over a four dimensional symplectic manifold. The notion of contact geometry plays a crucial role in the construction and we suggest a generalization of the instanton equations to five dimensional contact manifolds. Our main result is a calculation of the full perturbative partition function on a five sphere for the twisted supersymmetric Yang-Mills theory with different Chern-Simons couplings. The final answer is given in terms of a matrix model. Our construction admits generalizations to higher dimensional contact manifolds. This work is inspired by the work of Baulieu-Losev-Nekrasov from the mid 90's, and in a way it is covariantization of their ideas for a contact manifold.Comment: 28 pages; v2: minor mistake corrected; v3: matches published versio

Instanton on toric singularities and black hole countings

We compute the instanton partition function for ${\cal N}=4$ U(N) gauge theories living on toric varieties, mainly of type $\R^4/\Gamma_{p,q}$ including $A_{p-1}$ or O_{\PP_1}(-p) surfaces. The results provide microscopic formulas for the partition functions of black holes made out of D4-D2-D0 bound states wrapping four-dimensional toric varieties inside a Calabi-Yau. The partition function gets contributions from regular and fractional instantons. Regular instantons are described in terms of symmetric products of the four-dimensional variety. Fractional instantons are built out of elementary self-dual connections with no moduli carrying non-trivial fluxes along the exceptional cycles of the variety. The fractional instanton contribution agrees with recent results based on 2d SYM analysis. The partition function, in the large charge limit, reproduces the supergravity macroscopic formulae for the D4-D2-D0 black hole entropy.Comment: 29 pages, 3 fig Section 5 is improved by the inclusion of a detailed comparison between the instanton partition function and the D4-D2-D0 black hole entropy formula coming from supergravit

On Combinatorial Expansions of Conformal Blocks

In a recent paper (arXiv:0906.3219) the representation of Nekrasov partition function in terms of nontrivial two-dimensional conformal field theory has been suggested. For non-vanishing value of the deformation parameter \epsilon=\epsilon_1+\epsilon_2 the instanton partition function is identified with a conformal block of Liouville theory with the central charge c = 1+ 6\epsilon^2/\epsilon_1\epsilon_2. If reversed, this observation means that the universal part of conformal blocks, which is the same for all two-dimensional conformal theories with non-degenerate Virasoro representations, possesses a non-trivial decomposition into sum over sets of the Young diagrams, different from the natural decomposition studied in conformal field theory. We provide some details about this intriguing new development in the simplest case of the four-point correlation functions.Comment: 22 page

Thermodynamic limit of random partitions and dispersionless Toda hierarchy

We study the thermodynamic limit of random partition models for the instanton sum of 4D and 5D supersymmetric U(1) gauge theories deformed by some physical observables. The physical observables correspond to external potentials in the statistical model. The partition function is reformulated in terms of the density function of Maya diagrams. The thermodynamic limit is governed by a limit shape of Young diagrams associated with dominant terms in the partition function. The limit shape is characterized by a variational problem, which is further converted to a scalar-valued Riemann-Hilbert problem. This Riemann-Hilbert problem is solved with the aid of a complex curve, which may be thought of as the Seiberg-Witten curve of the deformed U(1) gauge theory. This solution of the Riemann-Hilbert problem is identified with a special solution of the dispersionless Toda hierarchy that satisfies a pair of generalized string equations. The generalized string equations for the 5D gauge theory are shown to be related to hidden symmetries of the statistical model. The prepotential and the Seiberg-Witten differential are also considered.Comment: latex2e using amsmath,amssymb,amsthm packages, 55 pages, no figure; (v2) typos correcte